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\title{Project 1 Tasks}
\author{Saurabh V. Pendse (ID: 001026185, Unity ID : svpendse)}
\date{\today}
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\begin{document}
\maketitle
\section{Task 1}
\begin{figure}[h]
        \centering
        \begin{subfigure}[b]{0.48\linewidth}
                \centering
                \includegraphics[width=\linewidth]{figure_task1_1k.eps}
                \caption{C = 1,000}
                \label{fig:task1_1k}
        \end{subfigure}
        \begin{subfigure}[b]{0.48\linewidth}
                \centering
                \includegraphics[width=\linewidth]{figure_task1_100k.eps}
                \caption{C = 100,000}
                \label{fig:task1_100k}
        \end{subfigure}
\caption{Results for Task 1}
\label{fig:task1}
\end{figure}

Figure \ref{fig:task1} shows the results for the Task 1.
We observe that the CLR is $0$ up to $\rho = 0.85$ in the $C = 1000$ case and up to $0.75$ in the $C = 100000$ case. For the $C = 1000$ case, we observe a sudden increase in the CLR value to about $0.025$ for $\rho = 0.95$. 
%Thus, there exists are threshold between $\rho = 0.85$ and $\rho = 0.95$ beyond which the CLR value shoots up due to the increased frequency of customer arrivals. 
We observe a very similar trend in the $C = 100000$ scenario. But the curve is much smoother and indicative of an exponential increase on account of more number of customers served, thereby increasing the instances of customers being lost beyond a certain value of $\rho$. In this case, we start observing an increase in the CLR from $\rho = 0.75$.

This increase is intuitive because as the value of $\rho$ increases, the inter-arrival times decrease, thereby increasing the frequency of arrival of the customers. This results in a higher probability of the queue being completely full, thereby increasing the customer loss rate.  Yes, these results were expected before running the experiment.

\section{Task 2}

\begin{figure}[h]
        \centering
        \begin{subfigure}[b]{0.48\linewidth}
                \centering
                \includegraphics[width=\linewidth]{figure_task2_1k.eps}
                \caption{C = 1,000}
                \label{fig:task2_1k}
        \end{subfigure}
        \begin{subfigure}[b]{0.48\linewidth}
                \centering
                \includegraphics[width=\linewidth]{figure_task2_100k.eps}
                \caption{C = 100,000}
                \label{fig:task2_100k}
        \end{subfigure}
\caption{Results for Task 2}
\label{fig:task2}
\end{figure}
Figure \ref{fig:task2} shows the results for the Task 2.
We observe that the CLR decreases with an increasing queue size. This behavior is consistent for both the $C =1000$ and $C = 100000$ cases. This behavior can be explained by the fact that as the queue size increases, and for a fixed $\rho = 0.85$ (thus a fixed inter-arrival time), there are reduced instances of customers being denied service due to a completely full queue. Thus, as the queue size increases, there are less and less number of customers denied service, thereby exponentially reducing the Customer Loss Rate (CLR) with increasing $K$. We observe that this behavior is much smoother for the $C = 100000$ case than for the $C = 1000$ due to the longer simulation time for the former case, which results in a more accurate reflection of the CLR as $C \longrightarrow \infty$.

\section{Task 3}

\begin{figure}[h]
\centering
\includegraphics[width=0.5\linewidth]{figure_task3.eps}
\caption{Results for Task 3}
\label{fig:task3}        
\end{figure}

Figure \ref{fig:task3} shows the results for the Task 3.
The analytical values are close to the experimentally observed values. In general, we observe an exponential increase in the Customer Loss Rate (CLR) with increasing value of $\rho$. This can be explained by the fact that as $\rho$ increases, $\lambda$ also increases (since $\mu = 1$). This means that the inter-arrival times decrease with the service times remaining more or less unchanged (since $\mu = 1$ is fixed). This results in an increased queue occupancy of $k$ or near to $k$ which leads to a greater number of customers being lost because of the queue being fully occupied. Thus , the Customer Loss Rate (CLR) is inversely proportional to the inter-arrival times. Now since the inter-arrival times follow an exponential distribution, we see an exponential increase int the CLR with increasing $\rho$ (i.e. decreasing inter-arrival times).

\section{Task 4}

\begin{figure}[h]
\centering
\includegraphics[width=0.5\linewidth]{figure_task4.eps}
\caption{Results for Task 4}
\label{fig:task4}        
\end{figure}

Figure \ref{fig:task4} shows the results for the Task 4.
We observe that the average waiting time $\overline{W}$ increases expotnentially with the parameter $\rho$ for $K = 100$.
This is because for a fixed queue size, as the value of $\rho$ increases, the inter-arrival times decrease. This means that customers arrive more frequently which increases the average occupancy of the queue. Thus every customer on an average has to wait for a longer period of time to get served. The waiting time can thus be said to be inversely proportional to the inter-arrival time. Since the inter-arrival times are exponentially distributed, we observe an exponential increase in the average waiting time $\overline{W}$ metric with increasing $\rho$ (i.e. decreasing inter-arrival times). Yes, the result is as expected.

\section{Task 5}

\begin{figure}[h]
\centering
\includegraphics[width=0.5\linewidth]{figure_task5.eps}
\caption{Results for Task 5}
\label{fig:task5}        
\end{figure}
Figure \ref{fig:task5} shows the results for the Task 5.
We observe an increasing trend of the simulation running time with increasing values of $\rho$ for a constant $K = 40$ and $C = 100,000$. 
%The fluctuations in some of the intermediate timing results are due to the experimental errors. These can be dealt with using multiple simulation runs to determine the average times for each run and the standard deviations. 
This behavior can be explained by the fact that as the value of $\rho$ increases, the inter-arrival times reduce, thereby increasing the frequency of arrivals. This leads to an increased average occupancy of the queue, with more number of customers being lost due to the queue being completely occupied. This effect is evident in the Task 3 (see Figure \ref{fig:task3}). This results in a longer simulation time in order to serve $C = 100,000$ customers (due to the greater number of losses). Yes, this was expected before running the experiment.

\newpage
\section{Task 6}
\begin{enumerate}
\item \textbf{What is "anticipatory flow state" and how is it related to routing asymmetry?}

Anticipatory flow state is a mechanism by which routers infer network conditions from the pattern of packet arrivals they observe. Routers establish such a state when they observe a new flow from A to B that is likely to generate a return flow from B to A. Routing asymmetry complicates the anticipatory flow state since the flow is no longer symmetric implying that a return flow might not always occur. This makes it difficult for routers to estimate network conditions from the pattern of packet arrivals.

\item \textbf{What is "fluttering" and what are three problems that it creates?}

Fluttering refers to rapidly-oscillating routing, wherein every alternate packet from a source travels via a different path. It creates the following problems : 
(a) A fluttering network path presents the difficulties that arise from unstable network paths, since one of the goals of the Internet architecture is to have stable network paths, (b) If the fluttering only occurs in one direction, then the path suffers from the problems of asymmetry. i.e. constructing reliable estimates of the path characteristics 
%such as round-trip time and available bandwidth, 
becomes potentially very difficult, since in fact there may be two different sets of values to estimate. (c) When two routes have different propagation times, then TCP packets arriving at the destination out of order can lead to spurious "fast retransmissions" by generating duplicate acknowledgements, wasting bandwidth.

\item \textbf{Are routing pathologies getting better or worse in the Internet? What arguments does the author provide for and against inferring trends from the data he presents?}

The author first mentions that experimental studies show that during 1995, the likelihood of a  user encountering a serious end-to-end routing problem more than doubled, and in 1996 was 1 in 30. From this result, the author argues that there is a trend showing that the Internet might get unstable  and network service will degrade to unacceptable levels in the future. He supports this argument using data indicating increased AS routing instability during the second quarter of 1996.

However, he also presents a contrasting view stating that it is dangerous to infer a trend from only two points. He makes an argument that 1995 was an atypical year for Internet stability; due to the transition from the NSFNET backbone to the commercially-operated backbone, and so it might not be a trend.

\item \textbf{Describe the organization of the paper. Is the structure what you would expect to see in a paper that deals with measurements? Explain.}

The paper starts by giving a brief overview of the purpose and the contribution. The next Sections (2 and 3) give overviews of related research. Sections 4 and 5 discuss the experimental and statistical methodology used for the analysis, the participating sites and the raw data. A number of routing pathologies are classified in Section 6. Sections 7 and 8 investigate routing stability and symmetry after removing the pathologies and studying the remaining measurements. Finally, Section 9 summarizes the findings of the research.

In my opinion, the organization does confirm to a paper that deals with measurements. Such a paper should have an extensive discussion about the results obtained upon analyzing the data, taking into account various constraints, relationships with a view to quantifying and discussing several aspects of the Internet service and the related implications.  
\end{enumerate}
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